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arXiv:cond-mat/0110169 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Quantum decay rates for driven barrier potentials in the strong friction limit

Joachim Ankerhold

Published Nov 26, 2001• 18 citations
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Abstract

Quantum decay rates for barrier potentials driven by external stochastic and periodic forces in the strong damping regime are studied. Based on the quantum Smoluchowski equation derived recently by Ankerhold, Pechukas, and Grabert [Phys. Rev. Lett. 87, 086802 (2001)] explicit analytical and numerical results are presented for the case of the resonant activation phenomenon in a bistable potential and the escape from a metastable well with oscillating barrier, respectively. The significant impact of quantum fluctuations is revealed.

DOI
10.1103/physreve.64.060102

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Quantum decay rates for barrier potentials driven by external stochastic and periodic forces in the strong damping regime are studied. Based on the quantum Smoluchowski equation derived recently by Ankerhold, Pechukas, and Grabert [Phys. Rev. Lett. 87, 086802 (2001)] explicit analytical and numerical results are presented for the case of the resonant activation phenomenon in a bistable potential and the escape from a metastable well with oscillating barrier, respectively. The significant impact of quantum fluctuations is revealed.
In the large damping limit we derive a Fokker-Planck equation in configuration space (the so-called Smoluchowski equation) describing a Brownian particle immersed into a thermal environment and subjected to a nonlinear external force. We quantize this stochastic system and survey the problem of escape over a double-well potential barrier. Our finding is that the quantum Kramers rate does not depend on the friction coefficient at low temperatures; i.e., we predict a superfluidity phenomenon in overdamped open systems. Moreover, at zero temperature we show that the quantum escape rate does not vanish in the strong friction regime. This result, therefore, is in contrast with the work by Ankerhold et al. [Phys. Rev. Lett. 87, 086802 (2001)]] in which no quantum tunneling is predicted at zero temperature.
The strong-friction regime at low temperatures is analyzed systematically starting from the formally exact path-integral expression for the reduced dynamics. This quantum Smoluchowski regime allows for a type of semiclassical treatment in the inverse friction strength so that higher-order quantum corrections to the original quantum Smoluchowski equation [J. Ankerhold, P. Pechukas, and H. Grabert, Phys. Rev. Lett. 87, 086802 (2001); J. Ankerhold and H. Grabert, Phys. Rev. Lett. 101, 119903 (2008)] can be derived. Drift and diffusion coefficients are determined by the equilibrium distribution in position and are directly related to the corresponding action of extremal paths and fluctuations around them. It is shown that the inclusion of higher-order corrections reproduces the quantum enhancement above crossover for the decay rate out of a metastable well exactly.
Based on the dynamical quantization method we derive a quantum phase-space non-Markovian Smoluchowski equation describing the non-inertial Brownian motion of a harmonic oscillator immersed in a generic environment. In the long-time regime we investigate the tunneling phenomenon by evaluating the quantum Kramers escape rate of a Brownian particle over a potential barrier. As far as a quantum thermal reservoir is concerned, it is found that our steady quantum Kramers rate may depend upon no friction constant in the strong friction domain. This preposterous feature may suggest the existence of non-dissipative quantum tunneling at the low-temperature range, including the zero temperature case. Lastly, we predict that our quantum escape rate non-exponentially decays on the edge of the breakdown of the dissipation-fluctuation relation.
We study the real time dynamics of a dissipative quantum system in a metastable state which may decay by crossing a potential barrier. Starting from an initial state where the system is in thermal equilibrium on one side of the barrier, the time evolution of the density matrix is evaluated analytically in the semiclassical approximation for coordinates near the barrier top. In a region about a critical temperature Tc{\mathrm{T}}_{\mathrm{c}}Tc​ large quantum fluctuations render the harmonic approximation of the potential insufficient and anharmonicities become essential. Accounting for non-Gaussian fluctuation modes, we show that the density matrix approaches a quasistationary state with a constant flux across the barrier. This extends our earlier results [Phys. Rev. E 51, 4267 (1995)] on the quantum generalization of the Kramers flux state to the region about Tc{\mathrm{T}}_{\mathrm{c}}Tc​. By matching the flux state onto the equilibrium state on one side of the barrier, we determine the decay rate out of the metastable state. The rate constant shows a changeover from thermally activated decay to quantum tunneling for temperatures below Tc{\mathrm{T}}_{\mathrm{c}}Tc​.
Using a generalized version of Langer's "imaginary part of the free energy" method the quantum decay rate Γ = A exp (−B) of a metastable ohmic oscillator — with harmonic frequency ω0 and quantum friction coefficient α — is studied for an almost symmetric barrier and a "free" exit space, involving an "inflection scattering" center. The unstable anomalous fluctuation mode (of the "breathing bounce" configuration in the partition function) is analytically continued in a fully "dynamical" manner. At zero friction and zero temperature the result for Γ agrees with the escape rate found from the pertinent Schrodinger equation by means of an extended WKB "outgoing waves" analysis. At slightly elevated temperatures — removing the earlier restrictions 12B exp (− ħω0/2k B T) → 0 and 2πk B T/ħω0 → 0 arising from the "two-state" and "sudden-flip" approximations — in the weak quantum damping regime (α ≲ 1) the decay rate shows a characteristic transition from quantal to thermal dependence on the bias energy ε. In the strong quantum damping regime (α ≳ 1) this dependence is always thermal while — in that case — at zero bias the oscillator becomes "dissipatively frozen" at zero temperature.